When doing the inclusion-exclusion principle, there are $n$ sets $A_1,\dots,A_n$, and we compute the sums of all $k$-ways intersections of these sets. Namely, for each $k\in \{1,\dots,n\}$, we compute $$ S_k=\sum_{i_1<\dots<i_k} |A_{i_1}\cap \dots \cap A_{i_k}| $$ The size of the union is then $S_1-S_2+S_3-\dots+(-1)^{n-1}S_n$. My question:
Is it always true that $S_1\ge S_2 \ge \dots \ge S_n$?
This seems to be true in practice. For example, if the universe is the set of permutations of an $n$-element set, and $A_i$ is the set of permutations which fix the $i^\text{th}$ element, then $$ S_k=\binom nk \cdot (n-k)!=\frac{n!}{k!}, $$ which is clearly weakly decreasing as a function of $k$.
These terms are definitely not decreasing in general. For a simple counterexample, consider when $A_1=\cdots=A_n=A$ is the entire set.